Finite Group Databasebeta

← #582575

Subgroups — #582575

Order 512.

2532 conjugacy classes of subgroups.

The subgroup lattice has 699 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#14668C212yesmaximal, complemented
3#14643C214yesmaximal, complemented
4#14627C212yesmaximal, complemented
5#14577C212yesmaximal, complemented
6#14467C212yesmaximal, complemented
7#14421C214yesmaximal
8#14264C212yesmaximal, complemented
9#14130C212yesmaximal, complemented
10#14053C212yesmaximal, complemented
11#13993C212yesmaximal, complemented
12#13974C212yesmaximal, complemented
13#13933C212yesmaximal, complemented
14#13872C211yesmaximal, characteristic
15#13839C211yesmaximal, characteristic
16C22×C4⋊Q8C2211yescharacteristic
17C22×#124C2212yes—
18C22×C4×D4C2211yescharacteristic
19C22×#117C2211yescharacteristic
20C23×C4⋊C4C2211yescharacteristic
21#2547C4116yescomplemented
22C2×#397C2212yes—
23C2×#397C2212yes—
24C2×#391C2214yes—
25C2×#391C2212yes—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#389C2212yes—
29C2×#388C2214yes—
30C2×#386C2212yes—
31C2×#385C2214yes—
32C2×#380C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2212yes—
36C2×C4×C4⋊C4C2212yes—
37C2×C4×C4⋊C4C2211yescharacteristic
38C2×C4×C22⋊C4C2212yes—
39C2×C4×C22⋊C4C2211yescharacteristic
40#1446C2214yes—
41#1444C2214yes—
42#1444C2214yes—
43#1443C2214yes—
44#1443C2214yes—
45#1442C2214yes—
46#1442C2214yes—
47#1442C2214yes—
48#1442C2214yes—
49#1441C2211yescharacteristic
50#1441C2211yescharacteristic
51#1441C2214yes—
52#1441C2212yes—
53#1440C2212yes—
54#1440C2214yes—
55#1439C2211yescharacteristic
56#1439C2211yescharacteristic
57#1439C2212yes—
58#1439C2214yes—
59#1439C2214yes—
60#1439C2214yes—
61#1438C2212yes—
62#1437C2214yes—
63#1436C2214yes—
64#1436C2214yes—
65#1436C2214yes—
66C4⋊#95C2212yes—
67C4⋊#95C2212yes—
68C4⋊#95C2212yes—
69C4⋊#95C2214yes—
70C4⋊#95C2212yes—
71C4⋊#95C2212yes—
72#1434C2214yes—
73C22⋊#95C2212yes—
74C22⋊#95C2214yes—
75C4×#95C2212yes—
76C23×Q8C2312yes—
77C23×D4C2311yescharacteristic
78C24×C4C2311yescharacteristic
79C24×C4C2311yescharacteristic
80C24×C4C2312yes—
81C2×C4⋊Q8—42no—
82C2×C4⋊Q8—42no—
83C2×C4⋊Q8C2×C418yescomplemented
84C2×#124—44no—
85C2×#124—44no—
86C2×#124C2×C4116yes—
87C2×C4×D4—42no—
88C2×C4×D4—42no—
89C2×C4×D4C2×C418yes—
90C2×#117—44no—
91C2×#117—42no—
92C22×C4⋊C4—44no—
93C22×C4⋊C4—42no—
94C22×C4⋊C4C2312yes—
95C22×C4⋊C4C2312yes—
96C22×C4⋊C4C2312yes—
97C22×C4⋊C4C2312yes—
98C22×C4⋊C4C2314yes—
99C22×C4⋊C4C2312yes—
100C22×C4⋊C4C2311yescharacteristic
101C22×C4⋊C4C2311yescharacteristic
102C22×C4⋊C4C2314yes—
103C22×C4⋊C4C2314yes—
104C22×C4⋊C4C2312yes—
105C22×C4⋊C4C2311yescharacteristic
106C22×C22⋊C4C2312yes—
107C22×C22⋊C4C2314yes—
108C22×C22⋊C4C2312yes—
109C22×C22⋊C4C2312yes—
110C22×C22⋊C4C2312yes—
111C22×C22⋊C4C2312yes—
112C22×C22⋊C4C2314yes—
113C22×C22⋊C4C2311yescharacteristic
114C22×C22⋊C4C2311yescharacteristic
115C22×C22⋊C4C2314yes—
116C22×C42C2312yes—
117C22×C42C2312yes—
118C22×C42C2311yescharacteristic
119C22×C42C2314yes—
120C22×C42C2312yes—
121#397C2×C4116yes—
122#397C2×C4116yescomplemented
123#391—48no—
124#391C2×C4116yescomplemented
125#389—44no—
126#389—44no—
127#389—44no—
128#388—48no—
129#386—44no—
130#385—48no—
131#380—44no—
132#380—42no—
133#380C2×C4116yes—
134#380C2×C4116yescomplemented
135C4×C4⋊C4—44no—
136C4×C4⋊C4C2×C418yescomplemented
137C4×C22⋊C4—44no—
138C4×C22⋊C4—42no—
139C2×#95C2312yes—
140C2×#95C2312yes—
141C2×#95C2312yes—
142C2×#95C2314yes—
143C2×#95C2312yes—
144C2×#95C2314yes—
145C2×#95C2314yes—
146C2×#95C2314yes—
147C2×#95C2314yes—
148C2×#95C2314yes—
149C2×#95C2314yes—
150C2×#95C2314yes—
151C2×#95C2314yes—
152C2×#95C2314yes—
153C2×#95C2314yes—
154C2×#95C2314yes—
155C2×#95C2314yes—
156C2×#95C2314yes—
157C2×#95C2311yescharacteristic
158C2×#95C2312yes—
159C2×#95C2312yes—
160C2×#95C2314yes—
161C2×#95C2311yescharacteristic
162C2×#95C2312yes—
163C2×#95C2312yes—
164C2×#95C2314yes—
165C2×#95C2314yes—
166C25C2411yescharacteristic
167C25C2411yescharacteristic
168C22×Q8—24no—
169C22×Q8—24no—
170C22×Q8—42no—
171C22×Q8—42no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C22×C418yes—
175C22×D4—22no—
176C22×D4—22no—
177C22×D4—22no—
178C22×D4—22no—
179C22×D4—41no—
180C22×D4—41no—
181C22×D4C22×C414yes—
182C23×C4—22no—
183C23×C4—24no—
184C23×C4—22no—
185C23×C4—24no—
186C23×C4—22no—
187C23×C4—22no—
188C23×C4—24no—
189C23×C4—24no—
190C23×C4—24no—
191C23×C4—44no—
192C23×C4—42no—
193C23×C4—44no—
194C23×C4—42no—
195C23×C4C22×C414yes—
196C23×C4C22×C414yes—
197C23×C4C2412yes—
198C23×C4C2414yes—
199C23×C4C2412yes—
200C23×C4C2412yes—
201C23×C4C2414yes—
202C23×C4C2414yes—
203C23×C4C2414yes—
204C23×C4C2411yescharacteristic
205C23×C4C2411yescharacteristic
206C23×C4C2411yescharacteristic
207C23×C4C2412yes—
208C23×C4C2411yescharacteristic
209C23×C4C2411yescharacteristic
210C4⋊Q8—416no—
211#124—432no—
212C4×D4—416no—
213#117—416no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—24no—
218C2×C4⋊C4—24no—
219C2×C4⋊C4—24no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—24no—
222C2×C4⋊C4—24no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—28no—
225C2×C4⋊C4—44no—
226C2×C4⋊C4—42no—
227C2×C4⋊C4—42no—
228C2×C4⋊C4—42no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—41no—
232C2×C4⋊C4—41no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—416no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—44no—
238C2×C4⋊C4—44no—
239C2×C4⋊C4—44no—
240C2×C4⋊C4—44no—
241C2×C4⋊C4—44no—
242C2×C4⋊C4—44no—
243C2×C4⋊C4—44no—
244C2×C4⋊C4C22×C418yes—
245C2×C4⋊C4C22×C418yes—
246C2×C4⋊C4C22×C414yes—
247C2×C4⋊C4C22×C418yes—
248C2×C22⋊C4—28no—
249C2×C22⋊C4—28no—
250C2×C22⋊C4—24no—
251C2×C22⋊C4—24no—
252C2×C22⋊C4—24no—
253C2×C22⋊C4—24no—
254C2×C22⋊C4—24no—
255C2×C22⋊C4—42no—
256C2×C22⋊C4—42no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—41no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—44no—
265C2×C22⋊C4—44no—
266C2×C22⋊C4—42no—
267C2×C22⋊C4—42no—
268C2×C22⋊C4—42no—
269C2×C22⋊C4—44no—
270C2×C22⋊C4—42no—
271C2×C22⋊C4—44no—
272C2×C22⋊C4—42no—
273C2×C22⋊C4—44no—
274C2×C22⋊C4C22×C4116yes—
275C2×C22⋊C4C22×C414yes—
276C2×C22⋊C4C22×C414yes—
277C2×C22⋊C4C22×C4116yes—
278C2×C42—22no—
279C2×C42—22no—
280C2×C42—24no—
281C2×C42—24no—
282C2×C42—24no—
283C2×C42—24no—
284C2×C42—24no—
285C2×C42—24no—
286C2×C42—42no—
287C2×C42—42no—
288C2×C42—44no—
289C2×C42—44no—
290C2×C42—44no—
291C2×C42C22×C414yes—
292C2×C42C22×C418yes—
293#95—28no—
294#95—24no—
295#95—24no—
296#95—24no—
297#95—44no—
298#95—42no—
299#95—44no—
300#95—44no—
301#95—41no—
302#95—41no—
303#95—42no—
304#95—42no—
305#95—44no—
306#95—42no—
307#95—44no—
308#95—44no—
309#95—44no—
310#95—44no—
311#95—44no—
312#95—44no—
313#95—44no—
314#95—44no—
315#95—44no—
316#95—44no—
317#95C22×C418yes—
318#95C22×C4116yes—
319#95C22×C4116yes—
320C24—21no—
321C24—22no—
322C24—21no—
323C24—21no—
324C24—22no—
325C24—22no—
326C24—21no—
327C24—22no—
328C24—22no—
329C24—21no—
330C24—21no—
331C24—21no—
332C24—21no—
333C24—21no—
334C24—22no—
335C24—22no—
336C24—21no—
337C24—22no—
338C24C23×C412yes—
339C24C23×C412yes—
340C24H(5,𝔽2)12yes—
341C24H(5,𝔽2)12yes—
342C24C2511yescharacteristic
343C2×Q8—216no—
344C2×Q8—48no—
345C2×Q8—48no—
346C2×Q8—84no—
347C2×Q8—84no—
348C2×Q8—84no—
349C2×Q8—84no—
350C2×D4—28no—
351C2×D4—28no—
352C2×D4—44no—
353C2×D4—44no—
354C2×D4—44no—
355C2×D4—44no—
356C2×D4—44no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—24no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—24no—
371C22×C4—24no—
372C22×C4—24no—
373C22×C4—24no—
374C22×C4—24no—
375C22×C4—24no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—24no—
384C22×C4—24no—
385C22×C4—22no—
386C22×C4—22no—
387C22×C4—24no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—24no—
391C22×C4—28no—
392C22×C4—28no—
393C22×C4—216no—
394C22×C4—216no—
395C22×C4—22no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—22no—
399C22×C4—22no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—24no—
405C22×C4—24no—
406C22×C4—24no—
407C22×C4—24no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—24no—
412C22×C4—24no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—24no—
425C22×C4—22no—
426C22×C4—24no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—44no—
431C22×C4—44no—
432C22×C4—48no—
433C22×C4—48no—
434C22×C4—44no—
435C22×C4—44no—
436C22×C4—48no—
437C22×C4—84no—
438C22×C4—88no—
439C22×C4—88no—
440C22×C4—84no—
441C22×C4C23×C412yes—
442C22×C4C23×C412yes—
443C22×C4C23×C418yes—
444C22×C4C23×C412yes—
445C22×C4C23×C418yes—
446C22×C4C23×C412yes—
447C22×C4C23×C412yes—
448C22×C4H(5,𝔽2)12yes—
449C22×C4H(5,𝔽2)12yes—
450C22×C4H(5,𝔽2)12yes—
451C22×C4H(5,𝔽2)14yes—
452C22×C42−1+414yes—
453C22×C42−1+412yes—
454C22×C42−1+414yes—
455C4⋊C4—216no—
456C4⋊C4—216no—
457C4⋊C4—44no—
458C4⋊C4—48no—
459C4⋊C4—48no—
460C4⋊C4—44no—
461C4⋊C4—44no—
462C4⋊C4—48no—
463C4⋊C4—48no—
464C4⋊C4—88no—
465C4⋊C4—88no—
466C4⋊C4—88no—
467C22⋊C4—232no—
468C22⋊C4—44no—
469C22⋊C4—44no—
470C22⋊C4—416no—
471C22⋊C4—48no—
472C22⋊C4—48no—
473C22⋊C4—48no—
474C22⋊C4—48no—
475C22⋊C4—84no—
476C22⋊C4—88no—
477C42—28no—
478C42—48no—
479C42—48no—
480C42—48no—
481C42—88no—
482C23—22no—
483C23—24no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—22no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—24no—
492C23—24no—
493C23—24no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—24no—
499C23—41no—
500C23—42no—
501C23—41no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23—41no—
506C23—41no—
507C23—42no—
508C23—42no—
509C23—41no—
510C23—42no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—42no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—41no—
522C23—42no—
523C23—41no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23#51814yes—
528C23C24×C411yescharacteristic
529C23C2×H(5,𝔽2)11yescharacteristic
530C23C2×H(5,𝔽2)11yescharacteristic
531C23C2×H(5,𝔽2)11yescharacteristic
532C23C2×H(5,𝔽2)11yescharacteristic
533C23C2×H(5,𝔽2)12yes—
534C23C2×H(5,𝔽2)11yescharacteristic
535C23C2×2−1+412yes—
536C23C2×2−1+411yescharacteristic
537C23C2×2−1+412yes—
538C23#58512yes—
539Q8—816no—
540D4—416no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—22no—
547C2×C4—22no—
548C2×C4—28no—
549C2×C4—28no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—28no—
553C2×C4—28no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—28no—
557C2×C4—24no—
558C2×C4—28no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—44no—
570C2×C4—44no—
571C2×C4—44no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—42no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—44no—
582C2×C4—42no—
583C2×C4—416no—
584C2×C4—416no—
585C2×C4—42no—
586C2×C4—42no—
587C2×C4—44no—
588C2×C4—42no—
589C2×C4—44no—
590C2×C4—42no—
591C2×C4—44no—
592C2×C4—42no—
593C2×C4—42no—
594C2×C4—42no—
595C2×C4—44no—
596C2×C4—42no—
597C2×C4—44no—
598C2×C4—42no—
599C2×C4—44no—
600C2×C4—42no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—44no—
604C2×C4—44no—
605C2×C4—42no—
606C2×C4—44no—
607C2×C4—44no—
608C2×C4—42no—
609C2×C4—42no—
610C2×C4—44no—
611C2×C4—42no—
612C2×C4—42no—
613C2×C4—42no—
614C2×C4—42no—
615C2×C4—42no—
616C2×C4—44no—
617C2×C4—44no—
618C2×C4—42no—
619C2×C4—44no—
620C2×C4—816no—
621C2×C4#51814yes—
622C2×C4#52014yes—
623C2×C4#52018yes—
624C2×C4H(3,𝔽4)14yes—
625C2×C4#56218yes—
626C2×C4#56314yes—
627C22—24no—
628C22—28no—
629C22—42no—
630C22—44no—
631C22—42no—
632C22—44no—
633C22—42no—
634C22—42no—
635C22—44no—
636C22—42no—
637C22—42no—
638C22—42no—
639C22—42no—
640C22—42no—
641C22—44no—
642C22—44no—
643C22—82no—
644C22—81no—
645C22—81no—
646C22—82no—
647C22—81no—
648C22—81no—
649C22C2×#51811yescharacteristic
650C22C2×#51811yescharacteristic
651C22C2×#52012yes—
652C22C2×#52011yescharacteristic
653C22C4×2−1+412yes—
654C22C2×H(3,𝔽4)11yescharacteristic
655C22C2×#56212yes—
656C22C2×#56311yescharacteristic
657C22#353611yescharacteristic
658C22#353611yescharacteristic
659C22#353612yes—
660C22#353711yescharacteristic
661C22#353711yescharacteristic
662C22#354112yes—
663C22#354212yes—
664C22#354412yes—
665C22#354612yes—
666C22#355012yes—
667C22#355011yescharacteristic
668C22#355111yescharacteristic
669C22#355212yes—
670C22#355512yes—
671C22#355912yes—
672C4—24no—
673C4—42no—
674C4—42no—
675C4—48no—
676C4—42no—
677C4—48no—
678C4—44no—
679C4—82no—
680C4—82no—
681C4—84no—
682C4—82no—
683C4—84no—
684C4#237918yes—
685C2—44no—
686C2—82no—
687C2C2×#237911yescharacteristic
688C2#3437911yescharacteristic
689C2#3441311yescharacteristic
690C2#3446012yes—
691C2#3448112yes—
692C2#5295811yescharacteristic
693C2#5296111yescharacteristic
694C2#5296312yes—
695C2#5296711yescharacteristic
696C2#5296811yescharacteristic
697C2#5309712yes—
subgroupG/H
center, Frattini, socle, cosocleC24C25
commutatorC23C24×C4
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #14668C2semidirect
3. #14643C2semidirect
4. #14627C2semidirect
5. #14577C2semidirect
6. #14467C2semidirect
8. #14264C2semidirect
9. #14130C2semidirect
10. #14053C2semidirect
11. #13993C2semidirect
12. #13974C2semidirect
13. #13933C2semidirect
21. #2547C4semidirect
83. C2×C4⋊Q8C2×C4semidirect
122. #397C2×C4semidirect
124. #391C2×C4semidirect
134. #380C2×C4semidirect
136. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#25472
#146682
#146432
#146272
#145772
#144672
#144212
#142642
#141302
#140532
#139932
#139742
#139332
#138722
#138392